Degree of a Vertex (Advanced Applications and Extensions)

  • 11th Grade
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| Questions: 20 | Updated: Dec 17, 2025
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1) In a directed graph, the degree of a vertex is the sum of:

Explanation

In digraphs, each vertex v has two measures:

in-degree deg⁻(v): edges entering v

out-degree deg⁺(v): edges leaving v

Total degree = deg⁻(v) + deg⁺(v).

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About This Quiz
Degree Of A Vertex (Advanced Applications And Extensions) - Quiz

Ready for more complex degree problems? In this quiz, you’ll work through situations involving multigraphs, weighted edges, and trickier structures. You’ll apply your foundational understanding in new ways, analyze special cases, and sharpen your problem-solving ability. With each step, you’ll deepen your grasp of how degree interacts with real graph... see morescenarios.
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2) If a directed edge connects vertex A to B, which statement is correct?

Explanation

The correct statement is B because a directed edge A → B leaves A and therefore contributes +1 to A’s out-degree while entering B and contributing +1 to B’s in-degree, illustrating the rule that each directed edge increases one vertex’s out-degree and another vertex’s in-degree, with total in-degrees and total out-degrees both equaling the number of edges.

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3) In a digraph with 6 vertices and 20 edges, the sum of all in-degrees is:

Explanation

The correct answer is 20 because in a digraph every directed edge contributes exactly one unit to some vertex’s in-degree, so the sum of all in-degrees equals |E| = 20, and by symmetry the total out-degree sum is also 20.

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4) A regular digraph with n vertices and constant out-degree r has how many edges?

Explanation

A regular digraph with n vertices and constant out-degree r has n × r edges because each of the n vertices emits exactly r outgoing edges, and since each edge is counted exactly once at its tail, multiplying n by r gives the total number of directed edges in the graph.

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5) A weighted directed graph represents traffic flow. At a junction v, in-flow = Σ (incoming weights) = 100 cars/hour; out-flow = Σ (outgoing weights) = 95 cars/hour. What does this imply?

Explanation

The correct implication is that the junction accumulates 5 cars per hour because the total in-flow of 100 cars/hour exceeds the out-flow of 95 cars/hour, meaning a net +5 cars/hour build up at that vertex, which reflects the flow-balance principle used in network modeling where excess in-flow indicates accumulation.

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6) If a graph has vertices of degrees (2, 2, 3, 3, 4, 4), which statements are true?

Explanation

Statements B and C are true because summing the degrees 2 + 2 + 3 + 3 + 4 + 4 yields 18, and dividing by 2 gives 9 edges according to the Handshaking Lemma, while statement A is false since a regular graph requires all degrees to be equal and statement D is false because there are actually four even-degree vertices (2, 2, 4, 4), not three.

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7) In any undirected weighted graph, if vertex v has incident edge weights 4, 6, and 10, the weighted degree of v is:

Explanation

The weighted degree of v is 20 because in an undirected weighted graph the weighted degree is defined as the sum of the weights of all edges incident to v, so adding 4, 6, and 10 gives 20, which represents the total cost or capacity associated with that vertex.

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8) Which relation must always hold for any digraph?

Explanation

Relation A must always hold because each directed edge increases exactly one vertex’s out-degree and exactly one vertex’s in-degree, meaning Σ deg⁺(v) = Σ deg⁻(v) = |E|, and none of the other listed relations consistently hold for all digraphs.

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9) In a directed graph, a loop adds 2 to the total degree of its vertex.

Explanation

True, because a loop in a directed or undirected setting simultaneously leaves and enters the same vertex, contributing +1 to in-degree and +1 to out-degree, giving a total degree contribution of 2.

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10) A digraph where every vertex has the same in-degree and out-degree is balanced.

Explanation

True, because a balanced digraph is defined by the condition deg⁺(v) = deg⁻(v) for every vertex v, representing systems with no accumulation or deficit of flow at any node.

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11) In any simple digraph with n vertices, maximum possible out-degree of a vertex is n.

Explanation

False, because in a simple digraph a vertex cannot have an edge to itself, so the maximum possible out-degree is n − 1 rather than n.

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12) If a vertex has out-degree 0 in a digraph, it’s called a sink.

Explanation

True, because a vertex with out-degree 0 emits no edges and only receives incoming edges, which is exactly the definition of a sink in directed graph terminology.

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13) If all vertices in a simple graph have even degree, the graph must be connected.

Explanation

False, because having all vertices of even degree does not guarantee connectivity—disconnected structures like several separate cycles can each have all-even degrees yet still form a disconnected graph.

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14) In any undirected graph, the sum of the degrees of all vertices is always an even number.

Explanation

True, because the sum of degrees in any undirected graph equals 2|E|, which is always an even number since twice any integer is even.

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15) In a directed graph, the total number of edges equals both the sum of all ________ and the sum of all ________

Explanation

In a directed graph the total number of edges equals both the sum of all out-degrees and the sum of all in-degrees because each edge contributes exactly one unit to the out-degree of its tail and one unit to the in-degree of its head, ensuring Σ deg⁺ = Σ deg⁻ = |E|.

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16) A vertex with in-degree 0 and out-degree > 0 is called a ________

Explanation

A vertex with in-degree 0 and out-degree greater than 0 is called a source because it emits edges but receives none, functioning as a starting point or origin in flow, dependency, or causal graphs.

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17) In an undirected graph, if the degrees of all vertices are 2, the graph consists of one or more ________

Explanation

If every vertex in an undirected graph has degree 2, the graph consists of one or more cycles because each vertex lies on exactly two edges, forcing the edges to connect in closed loops with no branching or endpoints.

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18) The maximum number of edges in a simple undirected graph with n vertices is ________.

Explanation

The maximum number of edges in a simple undirected graph with n vertices is n(n − 1)/2 because each vertex can connect to n − 1 distinct others but undirected edges are counted twice in that total, so dividing by 2 yields the true number of unique edges.

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19) If vertex A has degree 5 in a 6-vertex graph, it is adjacent to ________ other vertices.

Explanation

If vertex A has degree 5 in a 6-vertex graph, it is adjacent to 5 other vertices because degree counts the number of distinct neighbors in a simple graph, and the only vertex it cannot connect to is itself.

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20) A vertex with both in-degree and out-degree equal to 0 is called an ________ vertex.

Explanation

A vertex with both in-degree and out-degree equal to 0 is called an isolated vertex because it neither sends nor receives edges, making it disconnected from the rest of the digraph and giving it total degree 0.

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In a directed graph, the degree of a vertex is the sum of:
If a directed edge connects vertex A to B, which statement is correct?
In a digraph with 6 vertices and 20 edges, the sum of all in-degrees...
A regular digraph with n vertices and constant out-degree r has how...
A weighted directed graph represents traffic flow. At a junction v,...
If a graph has vertices of degrees (2, 2, 3, 3, 4, 4), which...
In any undirected weighted graph, if vertex v has incident edge...
Which relation must always hold for any digraph?
In a directed graph, a loop adds 2 to the total degree of its vertex.
A digraph where every vertex has the same in-degree and out-degree is...
In any simple digraph with n vertices, maximum possible out-degree of...
If a vertex has out-degree 0 in a digraph, it’s called a sink.
If all vertices in a simple graph have even degree, the graph must be...
In any undirected graph, the sum of the degrees of all vertices is...
In a directed graph, the total number of edges equals both the sum of...
A vertex with in-degree 0 and out-degree > 0 is called a ________
In an undirected graph, if the degrees of all vertices are 2, the...
The maximum number of edges in a simple undirected graph with n...
If vertex A has degree 5 in a 6-vertex graph, it is adjacent to...
A vertex with both in-degree and out-degree equal to 0 is called an...
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